For example, is saying a planar graph can always be coloured by 4 colours the same as saying a planar graph can never have more than 4 nodes that are all connected to each other?
There is a slight modification of the statement that may be correct; if you replace the condition of having a complete subgraph with that of having a complete minor, this is a very famous open problem called the Hadwiger conjecture. (It’s known to be true for the case n=5, i.e. every 5-chromatic graph has a K_5 minor.)
If a graph contains K_5 then it requires at least 5 colours to colour, but if a graph does not contain K_5 then it might still require 5 colours.
The example given by OscarCunningham is a great one.